Question: Solve for $x$ and $y$ using elimination. $\begin{align*}3x+9y &= -9 \\ -x-y &= 9\end{align*}$
We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $3$ $\begin{align*}3x+9y &= -9\\ -3x-3y &= 27\end{align*}$ Add the top and bottom equations. $6y = 18$ Divide both sides by $6$ and reduce as necessary. $y = 3$ Substitute $3$ for $y$ in the top equation. $3x+9( 3) = -9$ $3x+27 = -9$ $3x = -36$ $x = -12$ The solution is $\enspace x = -12, \enspace y = 3$.